R*D-fi43 123 FUNDRFIENTAL RESEARCH IN APPLIED #ATHEMATICS(U) MARYLAND L/i

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NFUNDAMENTAL RESEARCH IN APPLIED MATHEMATICS

(V)FINAL REPORT

IAUTHOR: F. W. J. OLVER

JUNE 12, 1984

U. S. ARMY RESEARCH OFFICE

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CONTRACT DAAG 29-80-C-0032

INSTITUTE FOR PHYSICAL SCIENCE AND TECHNOLOGY, UNIVERSITY OF MARYLAND

COLLEGE PARK, MD 20742

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Fundamental Research in Applied Mathematics Jan. 1, 1977 - Feb. 29, 1984

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(*) 6. CONTRACT OR GRANT NUMBER(e)

Frank W. J. Olver DAAG 29-77-G-0003* DAAG 29-80-C-0032

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University of MarylandCollege Park, MD 20742

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U. S. Army Research Office June 12,BE 1984AGPost Office Box 1221113NUBROPAE

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Unclassified

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Approved f or public release; distribution unlimited.

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119. SUPPLEMENTARY NOTES

The view, opinions, and/or findings contained in this report arethose of the author(s) and should not be construed as an officialDepartment of the Army position, policy, or decision, unless sordo4gtd TI y nthor f dn-mantntinn

* IS1. KEY 116"05(Continu on reverse side Of necessay OWd idevitify b block nmber)

Asymptotic analysis; computer arithmetic; difference equations; elementaryfunctions; elliptic functions; error analysis; extended-range arithmetic;floating-point system; Gaussian elimination; Legendre functions; level-indexarithmetic, Liouville-Green approximation; mathematical functions; multipleprecision; Newton's rule; ordinary differential equations; overflow; (continted)

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>-Problems were studied and solved in five main areas:asymptotic solutions of second-order differential equations; numericalsolution of difference equations; algorithms for the generation of mathema-tical functions; error analysis of numerical algorithms; computer arithmetic.

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19. Key Words (contlnued)polynomial evaluation; turning points; underflow; unrestricted algorithmsWhittaker functions; WKBJ approximation

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RESEARCH FINDINGS

I. Asymptotic solutions of second-order differential equations.

Publicationst: [1], [4], [51, [6], (16].

Three main problems in this category were studied and solved

successfully.

First, general connection formulas were obtained for the Liouville-

Green (or WKBJ) approximate solutions

{f(u,z) -1/4exp {±f f(u,z)}11/2dz}

of differential equations of the form

d 2w/dz - tu 2f(u,z)+g(u,z)}w , (1)

in which u is a large real or complex variable and the independent

variable z ranges over the complex plane. The differential equation may

have any number of turning points (zeros of f(u,z)) of arbitrary (includ-

ing fractional) multiplicity, and any number of singularities. There are

many potential physical applications of this work, including scattering

problems, transmission of radio waves, trapping of water waves and

hydrodynamic instability.

Second,asymptotic approximations were obtained for standard solu-

tions of Whittaker's form of the confluent hypergeometric equation:

d 2W [±4 T 2 Wdz z 2J

tReferences are listed on pages 6-8.

. .. %" Lq -'" " " W'

f or large positive values of the parameter pi that are uniform with

respect to unrestricted values of the argument z in the open interval

(0,oo) and bounded real values of the ratio ic/w. . The approximations are

in terms of parabolic cylinder functions, complete with error bounds.

This work closes one of the two important gaps in the asymptotic theory

of these frequently occurring functions in problems of mathematical

physics [251.

The third problem is the construction of uniform asymptotic solutions

of equation (1) in cases in which f(u,z) has a simple pole and a simple

turning point, the locations of which depend on a second parameter and may

coalesce for a critical value of that parameter. This is another major

unsolved problem in asymptotics that was described in [25]. J. J. Nestor,

a graduate student in the Applied Mathematics Program at the University of

Maryland has solved the problem fully for real variables, under the

direction of the principal investigator, and the results are given in

Nestor's Ph.D. thesis [16].

II. Numerical solution of difference equations.

In 1967 the principal investigator published a stable algorithm

for the computation of solutions of inhomogeneous linear difference equa-

4 tions of the second order [24]. This has since been incorporated in

various packages for computing special functions; see, for example, [191,

[22], [271-[31]. The extension of this algorithm to linear difference

equations of any order, or systems of such equations, was far from a

trivial problem. Working under the direction of the principal investigator,

D. W. Lozier solved this problem and was awarded a Ph.D. degree in the

Applied Mathematics Program at the University of Maryland for this work.

- . . . .. . . . . . . . . . . . . . . .* 77 4

3

So far Lozier's results have been issued as a U.S. Department of Commerce

i Report [20]; some of them have also been reproduced in a recent text [32].

*,e [Note. Lozier's work was not supported by the AROD since he was (and still

is) a full-time employee of the National Bureau of Standards. The

principal investigator's work on this problem was supported by AROD.]

III. Algorithms for the generation of mathematical functions.

Publications: [2], [3], [7], [11], [12], [13].

The construction of high-quality portable software for the genera-'9

tion of mathematical functions is an ongoing project at several universi-

ties and government laboratories around the world. The main contribLution

by the principal investigator and his co-workers, D. W. Lozier and J. M.

Smith, has been the construction of comprehensive packages for the

4 Legendre functions [7], [13], [21]. It was possible to cover extremely

large ranges of the parameters for these functions by the introduction of

r an "extended-range" arithmetic, in which a whole word is allotted to the

exponents of floating-point numbers, without undue sacrifice of speed.

The new packages have been incorporated in the GAMS library at the National

Bureau of Standards [22],and one is also available from the National

Technical Information Service [23].

A second type of problem that was studied is the construction of

unrestricted algorithms for the generation of mathematical functions, that

is, algorithms that will generate function values to any guaranteed

precision for any values of the variables. "The object here is twofold.

First, we seek, in effect, constructive numerical algorithms for the

functions which can then he used to produce day-to-day (restricted)

4, algorithms, based, for example, on expansions in series of Chebyshev

4

* polynomials or rational functions. Second, it was believed (correctly)

that new mathematical and computational tools would need to be developed

in the course of constructing the unrestricted algorithms, which might be

applicable to other computational problems.

So far, most of the elementary functions have been treated. An

unrestricted algorithm for the exponential function was described in 1 2],

[3]. Unrestricted algorithms for reciprocals, square roots, logarithms

and the trigonometric functions await testing at the National Bureau of

$ Standards, pending the development of a considerable extension to

R. P. Brent's multiple-precision package [17], [18].

A third area in the general study of algorithms for the generation

of mathematical functions consisted of work on some miscellaneous topics,

especially elliptic functions, that was initiated by B. C. Carlson during

his visit to Maryland in 1980-81; see [11], [12].

IV. Error analysis of numerical algorithms.

Publications: [8], (9], [10], [14].

In the course of the work on unrestricted algorithms described in

III, a new version of error analysis was developed, based on a logarithmic

definition of relative error, and applicable to real and complex arithmetic.

It was soon realized that this analysis might also be applicable to other

problems in numerical analysis. An application to Gaussian elimination

was made in [9] in collaboration with J. H. Wilkinson, and new explicit

error bounds of a posterlari type were obtained. Other successful

applications so far include polynomial evaluation and Newton's rule, and

this work is being readied for publication under the new AROD Contract

(DMAG 29-84-K-0022).

V. Computer arithmetic.

Publication: [15].

Overflow and underflow are continual problems in the construction

of comprehensive computing programs [26]. This is especially true for

robust algorithms for the generation of mathematical functions. It has

already been noted in III that a special form of computer arithmetic was

developed in connection with packages for Legendre functions, and this

eased considerably the problems of overflow and underflow in these cases.

The floating-point system (including extended-range arithmetic)

can be regarded as a mapping from the real line onto an R 2-space. The

principal investigator and his co-workers, C. W. Clenshaw and D. W. Lozier,

have been addressing the following problem. Are there other forms of

mapping onto an R -space in which the phenomena of overflow and underflow

are eliminated, in the sense that arithmetic operations or numbers that

lie outside the mapping essentially become trivial? An affirmative answer,

describing a number system based on iterated (or generalized) exponential

and logarithmic functions, was supplied recently in [15]. Since this

work was completed algorithms have been devised and tested for the new

arithmetic, called level-index arithmetic, and some test runs have also

been made at the National Bureau of Standards for generating Bessel and

Legendre functions. In consequence, level-index arithmetic is already a

feasible proposition for multiple-precision computations for which speed

is not an important consideration. Its feasibility for ordinary single-

and double-precision work may hinge upon the possibility of constructing

silicon chips to implement the arithmetic operations.

Ik76

Publications Supported by AROD Grant DAAG 29-77-G-0003and Contract DAAG 29-80-C-0032

[1] F. W. J. Olver, "General connection formulae for Liouville-Greenapproximations in the complex plane", Philos. Trans. Roy. Soc.London Ser. A 289 (1978), 501-548.

[2] C. W. Clenshaw and F. W. J. Olver, "An unrestricted algorithm forthe exponential function", SIAM J. Numer. Anal. 17 (1980), 310-331.

[3] F. W. J. Olver, "Unrestricted algorithms for generating elementaryfunctions", pp. 131-140, Fundamentals of Numerical Computation(Computer-Oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff,eds., Computing Suppl. 2, Springer, NY, 1980.

[4] F. W. J. Olver, "The general connection-formula problem for lineardifferential equations of the second order", pp. 317-343, SingularPerturbations and Asymptotics, R. E. Meyer and S. V. Parter, eds.,Academic Press, New York, 1980.

[5] F. W. J. Olver, "Asymptotic approximations and error bounds", SIAMRev. 22 (1980), 188-203.

[6] F. W. J. Olver, "Whittaker functions with both parameters large:uniform approximations in terms of parabolic cylinder functions",Proc. Roy. Soc. Edinburgh 86A (1980), 213-234.

[7] J. M. Smith, F. W. J. Olver, and D. W. Lozier, "Extended-rangearithmetic and normalized Legendre polynomials", ACM Trans. Math.Software 7 (1981), 93-105.

[8] F. W. J. Olver, "Further developments of rp and ap error analysis",IMA J. Numer. Anal. 2 (1982), 249-274.

[9] F. W. J. Olver and J. H. Wilkinson, "A posteriori error bounds forGaussian elimination", IMA J. Numer. Anal. 2 (1982), 377-406.

[101 F. W. J. Olver, "Error analysis of complex arithmetic", pp. 279-292,Computational Aspects of Complex Analysis, H. Werner, L. Wuytack,E. Ng and H. J. BUnger, eds., D. Reidel Publishing Company, Dordrecht,1983.

[11] B. C. Carlson and John Todd, "The degenerating behavior of ellipticfunctions", SIAM J. Numer. Anal. 20 (1983), 1120-1129.

[12] B. C. Carlson and John Todd, "Zolotarev's first problem - the bestapproximation by polynomials of degree < n-2 to xn - naxn '1 in[-1,1]", Aequationes Math. 26 (1983), 1-33.

[13] F. W. J. Olver and J. M. Smith, "Associated Legendre functions on thecut", J. Comp. Phys. 51 (1983), 502-518.

[14] F. W. 3. Olver, "Error bounds for arithmetic operations on computerswithout guard digits", IMA J. Numer. Anal. 3 (1983), 153-160.

7

,. [151 C. W. Clenshaw and F. W. J. Olver, "Beyond floating point", J. Assoc."5 Computing Machinery 31 (1984), 319-328.

[16] J. J. Nestor III, "Uniform asymptotic approximations of solutions ofsecond-order linear differential equations, with a coalescing simpleturning point and simple pole", Ph.D. Dissertation, Applied Mathema-tics Program, University of Maryland..A

.5- Other Referencesi'

. [17] R. P. Brent, "A Fortran multiple-precision arithmetic package", ACMTrans. Math. Software 4 (1978), 57-70.

[18] R. P. Brent, "Algorithm 524. MP, A Fortran multiple-precision arithme-tic package", ACM Trans. Math. Software 4 (1978), 71-81.

[19] W. J. Cody, "Preliminary report on software for the modified Besselfunctions of the first kind", Argonne National Lahoratory ReportTM-357 (1980) Argonne, Illinois, 33 pp.

(20] D. W. Lozier, "Numerical solutions of linear difference equations",Report NBSIR 80-1976, U. S. Department of Commerce, National Bureau

I-' of Standards, March 1980, 176 pp..

[21] D. W. Lozier and J. M. Smith, "Algorithm 567. Extended-range arithmeticand normalized Legendre polynomials", ACM Trans. Math. Software 7(1981), 141-146.

[22] National Bureau of Standards, Guide to available mathematical soft-ware , U. S. Department of Commerce Report NBSIR 84-2824 (1984).-

[23] National Technical Information Service, 5285 Port Royal Road,Springfield, VA 22161, Accession No. PB 82-250853.

[24] F. W. J. Olver, "Numerical solution of second-order linear differenceequations", J. Res. Nat. Bur. Standards 71B (1967), 111-129.

[25] F. W. J. Olver, "Unsolved problems in the asymptotic estimation of

special functions", pp. 99-142, Theory and application of specialfunctions, R. A. Askey, ed., Academic Press, New York, 1975.

(26] B. Parlett, ed., Special Issue on the Proposed IEEE Floating-pointStandard, ACM SIGNUM Newsletter (October 1979), 1-32.

[27] W. L. Sadowski and D. W. Lozier, "Use of Olver's algorithm to evaluatecertain definite integrals of plasma physics involving Chebyshevpolynomials", J. Comput. Phys. 10 (1972), 607-613.

[28] D. J. Sookne, "Bessel functions I and J of complex argument and integerorder", J. Res. Nat. Bur. Standards 77B (1973), 111-114.

(29] D. J. Sookne, "Certification of an algorithm for Bessel functions ofreal argument", J. Res. Nat. Bur. Standards 77B (1973), 115-124.

8

[30] D. J. Sookne, "Bessel functions of real argument and integer order",J. Res. Nat. Bur. Standards 77B (1973), 125-132.

[31] D. J. Sookne, "Certification of an algorithm for Bessel functions ofcomplex argument", J. Res. Nat. Bur. Standards 77B, 1973, 133-136.

[32] J. Wimp, Computation with recurrence relations, Pitman, Boston,1984.

:-.°.-

9

Personnel Support by AROD Grant DAAG 29-77-G-0003and Contract DAAG 29-80-C-0032

F. W. J. Olver (Principal Investigator),Institute for Physical Science and TechnologyUniversity of MarylandCollege Park, MD 20742 (1977-1984)

C. W. Clenshaw, Mathematics DepartmentCartmel CollegeUniversity of LancasterBailrigg, Lancaster, LAl 4YLENGLAND (1981)

B. C. Carlson, Department of Mathematics,

Iowa State UniversityAmes, IA 50010 (1980-1981)

J. J. Nestor, Department of Mathematics,University of MarylandCollege Park, MD 20742 (1984)

Note. It is expected that J. J. Nestor will be awarded a Ph.D. degreein the Applied Mathematics Program at the University of Marylandduring the summer of 1984. Part of his thesis work was supportedby the AROD contract.

THE VIEW, OPINIONS, AND/OR FINDINGS CONTAINED IN THIS REPORT ARE THOSE OFTHE AUTHOR AND SHOULD NOT BE CONSTRUED AS AN OFFICIAL DEPARTMENT OF THEARMY POSITION, POLICY, OR DECISION, UNLESS SO DESIGNATED BY OTHER DOCLMENTATION.

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